RegularPrime 2002-06-05 01:35:08 2005-04-20 02:26:14 Definition irregular prime % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here A prime $p$ is {\em regular} if the class number of the cyclotomic field $\mathbb{Q}(\zeta_p)$ is not divisible by $p$ (where $\zeta_p := e^{2 \pi i/p}$ denotes a primitive $p^\mathrm{th}$ root of unity). An {\em irregular prime} is a prime that is not regular. Regular primes rose to prominence as a result of Ernst Kummer's work in the 1850's on Fermat's Last Theorem. Kummer was able to prove Fermat's Last Theorem in the case where the exponent is a regular prime, a result that prior to Wiles's recent work was the only demonstration of Fermat's Last Theorem for a large class of exponents. In the course of this work Kummer also established the following numerical criterion for determining whether a prime is regular: \begin{itemize} \item $p$ is regular if and only if none of the numerators of the Bernoulli numbers $B_0$, $B_2$, $B_4, \ldots, B_{p-3}$ is a multiple of $p$. \end{itemize} Based on this criterion it is possible to give a heuristic argument that the regular primes have density $e^{-1/2}$ in the set of all primes~\cite{ir}. Despite this, there is no known proof that the set of regular primes is infinite, although it is known that there are infinitely many irregular primes. \begin{thebibliography}{9} \bibitem{ir} Kenneth Ireland \& Michael Rosen, \emph{A Classical Introduction to Modern Number Theory,} Springer-Verlag, New York, Second Edition, 1990. \end{thebibliography}